# Properties

 Label 287.2.c.a Level 287 Weight 2 Character orbit 287.c Analytic conductor 2.292 Analytic rank 0 Dimension 10 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$287 = 7 \cdot 41$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 287.c (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$2.29170653801$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( \beta_{1} - \beta_{7} ) q^{3} + ( 1 + \beta_{2} + \beta_{3} ) q^{4} + ( -1 - \beta_{3} - \beta_{4} ) q^{5} + ( -\beta_{1} + \beta_{5} - \beta_{7} + \beta_{8} ) q^{6} -\beta_{7} q^{7} + ( 1 + \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{8} + ( -1 + \beta_{2} - 2 \beta_{4} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( \beta_{1} - \beta_{7} ) q^{3} + ( 1 + \beta_{2} + \beta_{3} ) q^{4} + ( -1 - \beta_{3} - \beta_{4} ) q^{5} + ( -\beta_{1} + \beta_{5} - \beta_{7} + \beta_{8} ) q^{6} -\beta_{7} q^{7} + ( 1 + \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{8} + ( -1 + \beta_{2} - 2 \beta_{4} ) q^{9} + ( 1 - 2 \beta_{2} - \beta_{3} - \beta_{9} ) q^{10} + ( \beta_{5} + \beta_{6} + \beta_{7} ) q^{11} + ( \beta_{1} - 2 \beta_{6} + 2 \beta_{8} ) q^{12} + ( \beta_{1} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{13} + \beta_{8} q^{14} + ( -3 \beta_{1} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{15} + ( -2 + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{9} ) q^{16} + ( \beta_{1} - \beta_{6} + 3 \beta_{7} + \beta_{8} ) q^{17} + ( 1 + \beta_{3} + 2 \beta_{4} - 2 \beta_{9} ) q^{18} + ( -2 \beta_{1} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{19} + ( -2 - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{9} ) q^{20} + ( -1 - \beta_{4} ) q^{21} + ( \beta_{1} + \beta_{5} - 2 \beta_{7} - 2 \beta_{8} ) q^{22} + ( -1 + 3 \beta_{4} ) q^{23} + ( 3 \beta_{1} - \beta_{5} - 4 \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{24} + ( -1 + 2 \beta_{3} + 3 \beta_{4} - \beta_{9} ) q^{25} + ( -4 \beta_{1} + 3 \beta_{6} - \beta_{8} ) q^{26} + ( -\beta_{1} + \beta_{5} + 3 \beta_{7} + 3 \beta_{8} ) q^{27} + ( -\beta_{6} - \beta_{7} + \beta_{8} ) q^{28} + ( \beta_{1} + \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{29} + ( 3 \beta_{1} - 2 \beta_{5} + \beta_{6} - \beta_{7} - 4 \beta_{8} ) q^{30} + ( 3 - 2 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{9} ) q^{31} + ( 3 - \beta_{2} + 2 \beta_{9} ) q^{32} + ( -1 + 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{9} ) q^{33} + ( \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{34} + ( -\beta_{1} + \beta_{6} + \beta_{7} ) q^{35} + ( 2 - \beta_{3} - \beta_{4} ) q^{36} + ( -2 - \beta_{2} - 4 \beta_{3} - 3 \beta_{4} ) q^{37} + ( -\beta_{1} - 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} ) q^{38} + ( -5 + 4 \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{9} ) q^{39} + ( -5 - 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} ) q^{40} + ( -1 + 2 \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} + 2 \beta_{8} + \beta_{9} ) q^{41} + ( -1 - \beta_{2} + \beta_{4} - \beta_{9} ) q^{42} + ( 3 - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{9} ) q^{43} + ( \beta_{1} - \beta_{6} + 3 \beta_{7} ) q^{44} + ( 4 - 2 \beta_{2} + 2 \beta_{3} + 5 \beta_{4} - 3 \beta_{9} ) q^{45} + ( 3 - \beta_{2} - 3 \beta_{4} + 3 \beta_{9} ) q^{46} + ( 3 \beta_{1} - 2 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} + 3 \beta_{8} ) q^{47} + ( -3 \beta_{1} + 2 \beta_{5} - \beta_{6} - \beta_{7} + 3 \beta_{8} ) q^{48} - q^{49} + ( -1 + \beta_{2} + \beta_{3} - 3 \beta_{4} + 2 \beta_{9} ) q^{50} + ( 1 - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{51} + ( -\beta_{1} - 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} ) q^{52} + ( -\beta_{1} + 2 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} ) q^{53} + ( 3 \beta_{1} - 4 \beta_{6} - 8 \beta_{7} ) q^{54} + ( 3 \beta_{1} - \beta_{6} - 3 \beta_{7} - 2 \beta_{8} ) q^{55} + ( \beta_{1} - 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{56} + ( 6 + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{9} ) q^{57} + ( -\beta_{1} + 2 \beta_{5} + \beta_{6} - 5 \beta_{7} - 6 \beta_{8} ) q^{58} + ( 5 + 2 \beta_{2} + \beta_{3} - 3 \beta_{4} + 4 \beta_{9} ) q^{59} + ( -2 \beta_{1} - \beta_{5} + 3 \beta_{6} + 3 \beta_{7} - 4 \beta_{8} ) q^{60} + ( -3 + 3 \beta_{3} + 2 \beta_{4} - \beta_{9} ) q^{61} + ( -5 - \beta_{3} + 4 \beta_{4} + \beta_{9} ) q^{62} + ( -2 \beta_{1} + \beta_{7} + \beta_{8} ) q^{63} + ( 1 - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{64} + ( -5 \beta_{1} + 2 \beta_{5} + \beta_{6} + \beta_{7} + 4 \beta_{8} ) q^{65} + ( -3 + 2 \beta_{2} + \beta_{3} - 4 \beta_{4} + \beta_{9} ) q^{66} + ( -\beta_{1} - 2 \beta_{5} + 3 \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{67} + ( 2 \beta_{1} + \beta_{5} + \beta_{7} + \beta_{8} ) q^{68} + ( 2 \beta_{1} - 8 \beta_{7} - 3 \beta_{8} ) q^{69} + ( -\beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{70} + ( \beta_{5} + 3 \beta_{6} - 3 \beta_{7} - 6 \beta_{8} ) q^{71} + ( -1 + \beta_{2} - 3 \beta_{3} - 4 \beta_{4} + 3 \beta_{9} ) q^{72} + ( -1 - \beta_{3} - 2 \beta_{4} - 3 \beta_{9} ) q^{73} + ( 2 - 7 \beta_{2} - 5 \beta_{3} - \beta_{4} - 3 \beta_{9} ) q^{74} + ( 5 \beta_{1} - 3 \beta_{5} - 5 \beta_{6} - 4 \beta_{7} - 2 \beta_{8} ) q^{75} + ( -4 \beta_{1} - 2 \beta_{5} + 7 \beta_{6} + 2 \beta_{7} - 7 \beta_{8} ) q^{76} + ( 1 + \beta_{3} - \beta_{9} ) q^{77} + ( 7 + 6 \beta_{3} + 6 \beta_{4} - 2 \beta_{9} ) q^{78} + ( -2 \beta_{1} + 4 \beta_{5} - 4 \beta_{6} + 2 \beta_{8} ) q^{79} + ( 1 - 6 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{9} ) q^{80} + ( -2 \beta_{2} + \beta_{3} + 2 \beta_{4} - 4 \beta_{9} ) q^{81} + ( 8 - \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - 4 \beta_{7} + 3 \beta_{8} + \beta_{9} ) q^{82} + ( -3 + 4 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - \beta_{9} ) q^{83} + ( -2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{84} + ( \beta_{1} - \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{85} + ( -\beta_{2} - 2 \beta_{3} + 4 \beta_{4} - \beta_{9} ) q^{86} + ( -3 + 2 \beta_{2} + 5 \beta_{3} + 6 \beta_{4} - 3 \beta_{9} ) q^{87} + ( -2 \beta_{1} - \beta_{5} - \beta_{6} + 5 \beta_{7} + 2 \beta_{8} ) q^{88} + ( -2 \beta_{5} - \beta_{6} - 5 \beta_{7} + 3 \beta_{8} ) q^{89} + ( -5 + 4 \beta_{2} - 3 \beta_{3} - 9 \beta_{4} + 2 \beta_{9} ) q^{90} + ( -1 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{9} ) q^{91} + ( -4 + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} ) q^{92} + ( \beta_{1} + \beta_{5} + 4 \beta_{6} ) q^{93} + ( -5 \beta_{1} + \beta_{5} - 3 \beta_{6} - 8 \beta_{7} + \beta_{8} ) q^{94} + ( 3 \beta_{1} - \beta_{5} - 4 \beta_{6} - 4 \beta_{7} + 4 \beta_{8} ) q^{95} + ( 2 \beta_{1} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{96} + ( 5 \beta_{1} - 2 \beta_{5} - 4 \beta_{6} + 7 \beta_{7} + 2 \beta_{8} ) q^{97} -\beta_{2} q^{98} + ( 7 \beta_{1} - 2 \beta_{5} - 5 \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + 4q^{2} + 12q^{4} - 10q^{5} + 12q^{8} - 10q^{9} + O(q^{10})$$ $$10q + 4q^{2} + 12q^{4} - 10q^{5} + 12q^{8} - 10q^{9} + 8q^{10} - 16q^{16} + 20q^{18} - 22q^{20} - 12q^{21} - 4q^{23} - 4q^{25} + 14q^{31} + 18q^{32} - 2q^{33} + 20q^{36} - 22q^{37} - 46q^{39} - 58q^{40} - 4q^{41} - 8q^{42} + 18q^{43} + 50q^{45} + 8q^{46} - 10q^{49} - 22q^{50} + 14q^{51} + 62q^{57} + 34q^{59} - 28q^{61} - 44q^{62} + 8q^{64} - 36q^{66} - 20q^{72} + 12q^{74} + 12q^{77} + 78q^{78} - 4q^{80} + 10q^{81} + 74q^{82} - 20q^{83} - 6q^{84} + 12q^{86} - 8q^{87} - 54q^{90} - 14q^{91} - 30q^{92} - 4q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 13 x^{8} + 60 x^{6} + 118 x^{4} + 96 x^{2} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} + 5 \nu^{2} + 3$$ $$\beta_{4}$$ $$=$$ $$\nu^{6} + 7 \nu^{4} + 12 \nu^{2} + 5$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{9} - 13 \nu^{7} - 55 \nu^{5} - 78 \nu^{3} - 16 \nu$$$$)/5$$ $$\beta_{6}$$ $$=$$ $$($$$$2 \nu^{9} + 21 \nu^{7} + 70 \nu^{5} + 76 \nu^{3} + 17 \nu$$$$)/5$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{9} - 13 \nu^{7} - 55 \nu^{5} - 83 \nu^{3} - 36 \nu$$$$)/5$$ $$\beta_{8}$$ $$=$$ $$($$$$3 \nu^{9} + 34 \nu^{7} + 130 \nu^{5} + 189 \nu^{3} + 83 \nu$$$$)/5$$ $$\beta_{9}$$ $$=$$ $$\nu^{8} + 11 \nu^{6} + 40 \nu^{4} + 53 \nu^{2} + 19$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 3$$ $$\nu^{3}$$ $$=$$ $$-\beta_{7} + \beta_{5} - 4 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{3} - 5 \beta_{2} + 12$$ $$\nu^{5}$$ $$=$$ $$\beta_{8} + 7 \beta_{7} - \beta_{6} - 6 \beta_{5} + 18 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$\beta_{4} - 7 \beta_{3} + 23 \beta_{2} - 53$$ $$\nu^{7}$$ $$=$$ $$-8 \beta_{8} - 40 \beta_{7} + 7 \beta_{6} + 30 \beta_{5} - 83 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$\beta_{9} - 11 \beta_{4} + 37 \beta_{3} - 106 \beta_{2} + 243$$ $$\nu^{9}$$ $$=$$ $$49 \beta_{8} + 213 \beta_{7} - 36 \beta_{6} - 143 \beta_{5} + 385 \beta_{1}$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/287\mathbb{Z}\right)^\times$$.

 $$n$$ $$206$$ $$211$$ $$\chi(n)$$ $$1$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
204.1
 − 2.19548i 2.19548i − 2.10917i 2.10917i − 1.45275i 1.45275i − 1.06459i 1.06459i − 0.698160i 0.698160i
−1.82015 1.19548i 1.31294 −0.937607 2.17596i 1.00000i 1.25055 1.57082 1.70658
204.2 −1.82015 1.19548i 1.31294 −0.937607 2.17596i 1.00000i 1.25055 1.57082 1.70658
204.3 −1.44859 3.10917i 0.0984241 −3.65619 4.50392i 1.00000i 2.75461 −6.66693 5.29633
204.4 −1.44859 3.10917i 0.0984241 −3.65619 4.50392i 1.00000i 2.75461 −6.66693 5.29633
204.5 0.889527 2.45275i −1.20874 0.645522 2.18179i 1.00000i −2.85426 −3.01597 0.574209
204.6 0.889527 2.45275i −1.20874 0.645522 2.18179i 1.00000i −2.85426 −3.01597 0.574209
204.7 1.86664 0.0645923i 1.48436 1.44688 0.120571i 1.00000i −0.962521 2.99583 2.70081
204.8 1.86664 0.0645923i 1.48436 1.44688 0.120571i 1.00000i −0.962521 2.99583 2.70081
204.9 2.51257 1.69816i 4.31302 −2.49861 4.26675i 1.00000i 5.81162 0.116251 −6.27793
204.10 2.51257 1.69816i 4.31302 −2.49861 4.26675i 1.00000i 5.81162 0.116251 −6.27793
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 204.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
41.b Even 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{5} - 2 T_{2}^{4} - 6 T_{2}^{3} + 10 T_{2}^{2} + 9 T_{2} - 11$$ acting on $$S_{2}^{\mathrm{new}}(287, [\chi])$$.